L11a91

L11a90

L11a92

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Link Presentations

[edit Notes on L11a91's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -2, 11, -8, 9, -7, 8, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-10q^{3/2}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-12za^{3}-6a^{3}z^{-1}+z^{7}a+4z^{5}a+9z^{3}a+10za+5az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-4a^{6}z^{8}-13a^{4}z^{8}-19a^{2}z^{8}-10z^{8}-3a^{7}z^{7}-6a^{5}z^{7}-6a^{3}z^{7}-12az^{7}-9z^{7}a^{-1}-a^{8}z^{6}+5a^{6}z^{6}+27a^{4}z^{6}+38a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+29a^{5}z^{5}+50a^{3}z^{5}+45az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+4a^{6}z^{4}-10a^{4}z^{4}-19a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-8a^{7}z^{3}-34a^{5}z^{3}-60a^{3}z^{3}-47az^{3}-12z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-8a^{6}z^{2}-6a^{4}z^{2}-z^{2}a^{-2}+4a^{7}z+18a^{5}z+31a^{3}z+24az+7za^{-1}+a^{8}+3a^{6}+3a^{4}+a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-6a^{3}z^{-1}-5az^{-1}-2a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   7 1   6 2                 8 3     -5 0               12 7       5 -2             12 10         -2 -4           10 10           0 -6         8 12             4 -8       5 10               -5 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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